Remark 5.1 The evolution given in Eq. REF  involves several operations that are nonlocal in physical space, but local in Fourier space, including the convolutions with the operators \(P_H\) , its inverse, \(\widetilde{P}_0\)  and the second derivative associated to viscous diffusion (recall that the Fourier symbol of the second derivative in physical space is \(\mathcal {F}[\partial _x^2](k)=-(2\pi )^2k^2\) ). In this periodic framework, we will massively rely on the Discrete Fourier Transform (DFT) to evaluate the deterministic part at the RHS of Eq. REF . Corresponding available wavelengths are thus given by \(k=\lbrace -N/2+1,\dots ,0,\dots ,N/2\rbrace \Delta k\)  where \(\Delta k=1/L_{{\text{tot}}}\) . For full benefit of the Fast Fourier Transform (FFT) algorithm to evaluate the DFT, we choose \(N\)  to be a power of 2, i.e. \(N=2^n\) . Evaluations of the transfer operator \(\mathcal {L}\)  and the quadratic term proportional to the parameter \(\gamma \)  are performed in physical space, which implies several back and forth computations of the DFT and its inverse, in what is known as a pseudo-spectral method. Furthermore, to get rid of the aliasing error induced by the quadratic nonlinear term, we use a de-aliasing procedure based on the \(3/2\) -rule (see for instance Ref. [1]}).

